graph the linear inequality -2x-4y greater than or equal to sign 8

To graph the linear inequality \(-2x - 4y \geq 8\), follow these steps:

### Step 1: Rewrite the inequality in slope-intercept form
The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Let's first turn the given inequality into equality:

\[
-2x - 4y = 8
\]

Now, solve for \(y\):

\[
-4y = 2x + 8
\]

Divide both sides by \(-4\):

\[
y = -\frac{1}{2}x - 2
\]

Now we have the line equation \(y = -\frac{1}{2}x - 2\).

### Step 2: Graph the boundary line
Since the inequality includes "greater than or equal to" (\(\geq\)), the boundary line will be **solid**. This shows that points on the line are included in the solution set. The line has:
- Slope \(m = -\frac{1}{2}\)
- Y-intercept \(b = -2\)

To graph the line:
1. Plot the y-intercept (0, -2).
2. From that point, use the slope \(-\frac{1}{2}\), which means "down 1 unit" and "right 2 units," to find additional points on the line.
3. Draw the solid line through these points.

### Step 3: Shade the region
Since the inequality is \(-2x - 4y \geq 8\), which translates to \(y \geq -\frac{1}{2}x - 2\), you need to shade the region **above** the line because it includes values where \(y\) is greater than or equal to the expression on the right-hand side.

### Final Graph
1. Solid line for \(y = -\frac{1}{2}x - 2\).
2. Shade the region above the line.

This is the graph for the inequality \(-2x - 4y \geq 8\).

Would you like more details or need clarifications on any step?

Here are some follow-up questions:
1. What is the significance of the slope in a linear equation?
2. How does shading differ when using a "greater than" versus a "less than" inequality?
3. Can you explain how to determine whether to shade above or below a line?
4. How do you convert a standard form inequality into slope-intercept form?
5. What is the difference between solid and dashed boundary lines?

**Tip:** Always double-check if the inequality includes the boundary line (solid) or excludes it (dashed) to ensure correct shading.

Graph the linear inequality -2x - 4y greater than or equal to sign 8

Learn how to graph the linear inequality -2x - 4y >= 8 by converting it to slope-intercept form and graphing the boundary line. This solution covers key algebraic concepts and graphing techniques suitable for students in Grades 8-10.

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